Find roots of polynomial $$ p(x) = x^4-19x^3+90x^2-32x $$

The roots of polynomial $ p(x) $ are:

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 0.3864\\[1 em]x_3 &= 7.3551\\[1 em]x_4 &= 11.2585 \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x }$ from $ x^4-19x^3+90x^2-32x $ and solve two separate equations:

$$ \begin{aligned} x^4-19x^3+90x^2-32x & = 0\\[1 em] \color{blue}{ x }\cdot ( x^3-19x^2+90x-32 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ x^3-19x^2+90x-32 & = 0 \end{aligned} $$

One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.

Step 2:

Polynomial $ x^3-19x^2+90x-32 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

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